# 10: Pauli Spin Matrices

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We can represent the eigenstates for angular momentum of a spin-1/2 particle along each of the three spatial axes with column vectors:

\begin{aligned} &|+z\rangle=\left[\begin{array}{l} 1 \\ 0 \end{array}\right] \quad|+y\rangle=\left[\begin{array}{l} 1 / \sqrt{2} \\ i / \sqrt{2} \end{array}\right] \quad|+x\rangle=\left[\begin{array}{l} 1 / \sqrt{2} \\ 1 / \sqrt{2} \end{array}\right] \\ &|-z\rangle=\left[\begin{array}{l} 0 \\ 1 \end{array}\right] \quad|-y\rangle=\left[\begin{array}{l} i / \sqrt{2} \\ 1 / \sqrt{2} \end{array}\right] \quad|-x\rangle=\left[\begin{array}{r} 1 / \sqrt{2} \\ -1 / \sqrt{2} \end{array}\right] \end{aligned}\tag{10.1} \nonumber

Similarly, we can use matrices to represent the various spin operators.

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